Guidelines

Failure to follow this guidance might result in a penalty of up to 10% on your marks.

I.      Submit a single PDF document with questions in ascending order. This can be produced for example in Word, LaTeX or MATLAB Live Script. Explain in detail your reasoning for every mathematical step taken.

II.      Do not write down your name, or student number, or any information that might help identify you in any part of the coursework. Do not write your name or student number in the  title  of your  coursework  document file.  Do  not  copy  and  paste  the  coursework questions into your submission – simply rewrite information where necessary for the sake of your argument.

III.      Insert relevant graphs or figures, and describe any figures or tables in your document. All figures must be labelled, with their axes showing relevant parameters and units.

IV.      You will need MATLAB coding to solve some questions. Include all code as pasted text (for the purposes of plagiarism checks) in an Appendix at the end of your document. Remember to comment on your code, explaining your steps.

This coursework counts towards 12.5% of your final ENGF0004 grades and comprises two questions, referred to as models. Each model is worth a total of 100 marks, both making up 50% of your CW 2.

On Academic Integrity (Read more about it here)

Academic integrity means being transparent about our work.

•    Research: You are encouraged to research books and the internet. You can also include and paraphrase any solution steps accessible in the literature and online content if you reference them.

•   Acknowledge others: We are happy when you acknowledge someone else's work. You are encouraged to point out if you found inspiration or part of your answers in a book, article or teaching resource. Read more about how to reference someone else's work here and how to avoid plagiarism here.

•    Do  not share and do  not  copy: We expect students not to share and  not to copy assessment solutions or MATLAB code from their peers, even if partially.

•    Do  not  publish  ENGF0004  assessment  material:  We  expect  students not  to share ENGF0004 assessment materials on external online forums, including tutoring or "homework" help websites.

Students found in misconduct can receive a 0 mark in that assessment component and have a record of misconduct in their UCL student register. In some extreme cases,       academic misconduct will result in the termination of your student status at UCL.

Model 1: Iceberg Movement [50%]

Figure 1. Image of an iceberg.

Icebergs  impact the  ocean environments they  pass through ,  changing them through  melting  and releasing fresh water. Studying their movement is important not only for predicting their impact on ecosystems, but can also help plan vessel movement, understand the implications of global warming on ice melting and even evaluate the potential for using icebergs as a source of fresh water.

In this model, you will work as a researcher to understand the movement of an iceberg, study its temperature and evaluate the practical implementation of towing icebergs to use them as a source of fresh water.

Question 1 [20 marks]

Satellite imagery was used to find the density of the iceberg empirically matched the function

(, , ) = ( −  3 )

The volume of the iceberg has been found to approximate the region bound by the elliptical function  2  + 22  +  = 16, and the planes  = 0,  = 2,  = 0,  = 2.

a)  [12] Find the mass of the iceberg.

b)  [8] Propose a simple model to determine what proportion of the iceberg is submerged. Describe briefly the steps taken in developing your model. Discuss briefly if you can modify your model to find the height of the iceberg that is submerged.

To develop this model, use the knowledge that the weight of the iceberg and the buoyant force (weight of the fluid displaced by the iceberg) are in equilibrium.

Question 2 [20 marks]

The satellite images were used to find the following trajectory for the iceberg movement.

 () =  +  = (33  + 4) + (2  + 23 )

a)  [5]  Implement  in  MATLAB  a  model  that  tracks  the  position  of  the  iceberg  and characterise its movement. Discuss briefly which features are you going to find and why.

b)  [10] Calculate the work done by the current force acting on the iceberg to move it from its initial position at  = 0 to its position at  = 25.

c)   [5] Determine if this force is conservative.

Question 3 [20 marks]

After day 25, the iceberg entered a part of the ocean where the current is well studied and its force has been found to be

 = (2   − )

Where  is a known coefficient.

a)   [12] Find the work done if the iceberg travelled:

•    counter clockwise along a quarter-circle with an equation  2  +  2  = 1 from point (1,0) to point (0, 1).

•    in a straight line from point (1,0) to point (0, 1) .

b)  [8] Discuss what the implications of your findings in a) are in relation to a practical plan to tow an iceberg through this part of the ocean. Outline a proposal for such a plan.

Question 4 [20 marks]

The temperature  at a point (, ) through a horizontal cross-section of the iceberg is given by the function

(, ) = 

where  is measured in Celsius and  ,  in meters.

a)  [10] Find the gradient of  analytically. Plot a contour plot of  and a vector plot of the gradient of  in MATLAB. Explain what can be deduced from these plots. Intuitively, draw the direction in which the temperature increases the fastest at the point (2, 0) .

To  help  you  answer,  you  might  find  it  useful  to  find  the  temperature  and  the temperature gradient at some points of your choosing.

b)  [10] The directional derivative of  in the direction of the unit vector  is defined as

 = ∇ ⋅

Using your knowledge of vectors, show analytically in which direction  the directional derivative  is maximised at the point (2, 0), or in other words in which direction the temperature increases the fastest at this point. What is the maximum rate of increase at this point?

Question 5 [20 marks]

Researchers have worked to determine the statistical probabilities of encountering tabular icebergs (flat and approximately rectangular) of certain dimensions in this area of the ocean. They established that the average length  of an iceberg is 10m, while its average width  is 5m, where the probabilities for each dimension are modelled by the exponential probability density functions

1 () =   −/10

2 () =   −/5

 ≥ 0

 ≥ 0.

The probabilities for each dimension are independent.

a)  [12] Find the probability that the perimeter of an iceberg is less than 40m.

The researchers later moved on to investigate a new area. A previous study in this area, dating back 25 years, had found that the likelihood of an iceberg to measure less than 30m in length is 30%. In the first week of their investigation, the researchers encountered 10 icebergs, 6 of which measured above 30m.

b)  [8] Is it likely that the size distribution of icebergs in the area has changed? Do the researchers have reason to continue their investigation?

Model 2: Planning and control of Operations [50%]

Figure 2. An image of a microchip.

Managing and planning inventory (materials or finished goods) is key to manufacturers and retailers , as was apparent with the shortages of goods as a result of the global Covid- 19 pandemic. Inventory is essential to enabling companies to operate smoothly under unforeseen circumstances. A considerable amount of resources are invested in maintaining any inventory level, tied up in the value of the materials and goods themselves, but also  into the associated carrying costs (for example, storage,  utilities, salaries, opportunity costs, etc.). Companies cannot afford to engage money in excessive inventory, any of which would also have a higher associated carrying cost.

Hence, companies are interested in finding the optimal inventory level which allows them to maintain their operations without interruption while minimising the investment required.

In this model, you will investigate a number of problems relating to planning the operations of a company which produces microchips, servicing different industries, with the goal of proposing solutions which optimise their operations.

Question 1 [25m]

A  basic  model  which  aims  to  determine  the  quantity  of  inventory  which  minimises  its management cost is based on minimising the sum of the ordering costs and carrying costs. A number of simplifying assumptions are made:

•    Demand is known with certainty and is constant.

•    No shortages are allowed.

•    The order quantity is received at once.

Ordering cost (): the cost per order () is multiplied by the number of orders. Since demand  is known, the number of orders can be rewritten as / , where  is order size.

 = 

Carrying cost (): the carrying cost per unit (ℎ) is multiplied by the average inventory level given by /2.

 = 

The total inventory cost () can therefore be minimised for an optimal value of order quantity  :

 =  +  =  + 

a)  [12] Use MATLAB to explore the relationship between  ,   and  by plotting their curves as a function of  for set values of  ,  and ℎ . Use your plot to decide at what order quantity  the total cost  is minimised.

Choose plausible values for  ,  , ℎ and range for  by reasoning what their relative values would be. You might find it useful to try a few different sets of values, and the effect of changing one parameter.

b)  [13] The model discussed so far only considers the inventory cost for a single product. Analytically, develop a matrix model which can find the optimal order quantity   and total inventory cost  for a product  if the company produces  number of products. When presenting your model, show the elements of the matrices used.

Implement you model in MATLAB and find the optimal order quantity    and total inventory cost  for a product  if the company produces three products with cost per order, demand and carrying cost per unit as shown in Table 1.

When  developing this  matrix  model,  consider that for  each  product  the  equation

         ℎ   

would hold true, where   ,   , ℎ  are the cost per order, demand and carrying cost per unit for product  .

Table 1. Demand and costs associated with the three products the company manufactures.

Question 2 [15m]

The manufacturer produces three microchip products, for each of which there are three main manufacturing steps – deposition, lithography and ionization. The manufacturer’s factory is currently able to complete 70h of deposition, 181h of lithography and 41h of ionization per day.

Product 1 requires 8 hours of deposition, 5h of lithography and 1h of ionization. Product 2 needs 3 hours of deposition, 12h of lithography and 3h of ionization. Product 3 requires 1h of deposition, 10 of lithography and 2h of ionization.

a)  [10] Use LU-decomposition to find the optimal balance between production of products 1, 2 and 3 to maximise the use of all available work power.

b)  [5] Discuss the advantages of using the LU-decomposition method to solve this system in  comparison  to  solving  with  the  inverse  method  by  considering  the  types  of operations that need to be performed.

Question 3 [30m]

The demand for different types of microchips changes every following year. The manufacturer is interested to understand what the trend for the relative share of demand for each product is so that they can plan the expansion of their plant.

They hired market analysts who researched the industry and found out that the likelihood that an industry will continue to use one of the three products or switch to another after each year is given by the probability matrix :

From P1     From P2     From P3

Given this, the proportion of demand for each product from the total after, for example, two years can be found by   =   for an initial state   .

a)  [10] Find the eigenvalues and eigenvectors of this matrix analytically (without the use of in-built software tools).

b)  [10]  Discuss  the  implications  of  the  eigenvalue   = 1   and  its  corresponding eigenvector. Consider what this would imply for the state of the system at that stage.

You might find it useful to consider the relative demand for each product after 5, 15 and 25 year (  ,   and  ) for an initial state   :

0.25

0.55

Note, the elements of  should add up to 1.

c)   [10] Briefly consider and discuss how you would adjust the strategies discussed in Questions 1 and 2 to adapt to this new demand distribution between products, given the company aims to produce 3600 microchips in total.